Rule-based transformations for geometric modelling

TL;DR

This paper introduces a formal, rule-based approach to defining and ensuring the consistency of topology-based geometric models using labelled graphs and constraints.

Contribution

It proposes a novel formal framework that models topology-based geometric objects as labelled graphs with constraints to guarantee topological and embedding consistency.

Findings

Defines topology-based geometric objects as labelled graphs.

Ensures topological consistency through labelling constraints.

Provides a formal method for geometric modelling validation.

Abstract

The context of this paper is the use of formal methods for topology-based geometric modelling. Topology-based geometric modelling deals with objects of various dimensions and shapes. Usually, objects are defined by a graph-based topological data structure and by an embedding that associates each topological element (vertex, edge, face, etc.) with relevant data as their geometric shape (position, curve, surface, etc.) or application dedicated data (e.g. molecule concentration level in a biological context). We propose to define topology-based geometric objects as labelled graphs. The arc labelling defines the topological structure of the object whose topological consistency is then ensured by labelling constraints. Nodes have as many labels as there are different data kinds in the embedding. Labelling constraints ensure then that the embedding is consistent with the topological structure. Thus, topology-based geometric objects constitute a particular subclass of a category of labelled graphs in which nodes have multiple labels.